• Week 1
• Week 2
• Week 3
• Week 4
• Week 5
• Week 6
• Week 7
• Week 8
• Week 9
• Week 10
• Week 11
• Week 12
• Week 13

Here is the plan as I see it now, it is, of course, subject to revision as the start date of the course approaches.

Week 1 - The Discovery of Neutron Stars, Black Holes and Gamma-Ray Bursts

Problem 1 - The Eddington Luminosity

There is a natural limit to the luminosity a gravitationally bound object can emit. At this limit the inward gravitational force on a piece of material is balanced by the outgoing radiation pressure. Although this limiting luminosity, the Eddington luminosity, can be evaded in various ways, it can provide a useful (if not truly firm) estimate of the minimum mass of a particular source of radiation.

1. Consider ionized hydrogen gas. Each electron-proton pair has a mass more or less equal to the mass of the proton (m_p) and a cross section to radiation equal to the Thompson cross-section (σ_T).
2. The radiation pressure is given by outgoing radiation flux over the speed of light.
3. Equate the outgoing force due to radiation on the pair with the inward force of gravity on the pair.
4. Solve for the luminosity as a function of mass.

Problem 2 - Minimum Masses

The observations of Sco X-1 and the quasars 3C 48 and 3C 273 can give a lower limit on the mass of the sources if they are gravitationally bound.

The source discovered by Giacconi et al. is now known as Sco X-1.

1. What is the most likely distance to Sco X-1 given its location on the sky?
2. At this distance given the flux estimate in the Giacconi et al., what is the luminosity of Sco X-1?
3. What is the minimum mass of Sco X-1?

You can estimate distances to 3C 48 and 3C 273 using the redshift of the objects. At the time, the Hubble constant was thought to be around 100 km/s/Mpc. The conventional wisdom is that it is 72 km/s/Mpc. Feel free to use either value.

1. What are the distances to 3C 48 and 3C 273? Feel free to neglect cosmological effects (for bonus points use Ω_M = 0.3 and Ω_Λ=0.7).
2. Do your distances agree with those in the papers?
3. The paper gives estimates of the luminosity in the optical and radio of these sources. What are the minimum masses of the objects using the optical and radio luminosities?

Problem 3 - Gamma-ray Burst Energetics

About how much energy is released in a gamma-ray burst if they are

1. In low Earth orbit?
2. At the distance of the moon?
3. At the disance of Pluto?
4. Near the center of the galaxy?
5. At cosmological distances?

What are the Eddington limiting masses for each of these scenarios? In which scenarios, would you be surprised if a gamma-ray burst repeated?

Estimate the amount of energy released by the following events. Use the formula GMm/R.

1. A nuclear explosion - 5 kg of fissile plutonium. Each plutonium atom releases about 200 MeV as it splits.
2. The collision of two asteroids. Assume each is 10 km in radius with a density of 3 g cm^-3.
3. The collision of two neutron stars. Assume each is 10 km in radius with a density of 10¹⁵ g cm^-3.
4. The collision of an asteroid with a neutron star.

Week 2 - Rotation-Powered Neutron Stars

Problem 1 - Spinning Neutron Stars

We will estimate how quickly we would expect neutron stars to spin, how much energy is stored in their spin and other interesting facts about spinning neutron stars.

1. The sun rotates every 24-30 days depending on latitude. How quickly would it rotate if it were compressed to 10km in radius while conserving its angular momentum? Its current radius is 7 x 10¹⁰ cm.
2. How fast could a neutron star rotate without breaking up? Consider the neutron star to be 1.4 and have a radius of 10 km and compare the centripetal acceleration of a bit of material on the surface to the gravitational acceleration.
3. How much angular momentum and rotational energy does a neutron star have? Use

   I ≈ 0.21

   M R2 1-2 G M R c2

and a spin period of break-up, 1.6 ms, 33 ms and 6 s.

Problem 2 - Original Spin

If you know the age of a neutron star, its current period and its period derivative, you can estimate its original spin.

1. Using the results from the lectures, derive a formula for P₀ in terms of the age, current period and period derivative of a pulsar.
2. The table below lists pulsars that are associated with historical supernovae. Complete the table by calculating the original spin of these neutron stars.

   Name	Age [yr]	Period [s]	P-dot [10-15 s s-1]	P0 [s]
   B0531-21	949	0.0331	422.69
   B0540-69	1000	0.050	480
   B1951+32	64000	0.0395	5.8
   J0205+6449	822	0.06568	193

3. Why do we know the age of the first and last pulsars so accurately?

Week 3 - The Structure of Neutron Stars

Problem 1 - Newtonian Polytropes

1. Consider a star of mass, M, and radius, R. Construct by dimensional analysis a characteristic pressure and a characteristic density from these quantities and Newton's constant, G.
2. A polytropic equation of state is a power-law relationship between pressure and density, P = K ρ^α. Substitute the characteristic pressure and density into the polytropic equation of state to derive a mass-radius relation.
3. Which values of α have special properties? What are they?

Problem 2 - Central Pressures

We will calculate the central pressure of a incompressible star in Newtonian physics and general relativity.

1. Use the General Relativistic equations of hydrostatic equilibrium to determine the central pressure of a star of mass M and radius R. The material is incompressible, i.e. its density is constant. After integrating (9) from the center where the enclosed mass, u, vanishes, it is easiest to integrate (10) from the surface where the pressure vanishes. Write your answer as a function of M and R by eliminating the constant density from the result.
2. By dimensional analysis, figure out where the factors of G and c appear in the General Relativistic equations of hydrostatic equilibrium.
3. Derive the Newtonian equations of hydrostatic equilibrium by taking the limit of c → ∞.
4. Redo the pressure calculation in Newtonian physics.
5. By dimensional analysis, figure out where the factors of G and c appear in your answer to (1).
6. Check your answers by taking the limit of c → ∞ for your answer to (5) and comparing it with the answer to (4).
7. What is the minimal radius for a constant-density star of a given mass? What is the maximal mass for a star of a particular density? What is the maximal mass for a star at nuclear density, 10¹⁵ g cm^-3?

Problem 3 - Neutron Star Masses

Calculate from dimensional analysis the typical mass of a neutron star.

1. Use the characteristic density and pressure of a star that you derived in Problem 1. Neutron stars have relativistic neutrons so the pressure is about the density times c². Use this to derive a relationship between the mass and radius of the star.
2. A relativistic degenerate gas has a density of one particle in a cube a Compton wavelength on a side. Combine this with the result from Part 1 to solve for the mass of the star.

Week 4 - Neutron-Star Cooling and Heating

Problem 1 - Thermodynamics and General Relativity

In general relativity if two bodies are in thermodynamic equilibrium,

T1 1 + z1

=

T2 1 + z2.

We can exploit this relationship along with Kirchoff's law to derive some interesting facts about how light travels from a neutron star to our telescopes. You can experiment with neutron-star lensing with this Java Applet.

1. Because everything is in thermodynamic equilibrium, we can safely assume that the neutron star of mass M and radius R emits as a blackbody at a temperature T. Calculate the total power emitted from the neutron star surface in the frame of the neutron star surface.
2. Calculate the total power received at infinity. Let the redshift of the surface be z.
3. Let the space surrounding the star be filled with blackbody radiation in thermal equilibrium with the surface of the neutron star. You can imagine that the neutron star is in a gigantic thermos bottle. Let T_∞ be the temperature of this blackbody radiation measured at infinity (i.e. z=0). What is T_∞?

Now here comes Kirchoff's law: in thermodynamic equilibrium a body emits as much as it receives.

4. How much power does the neutron star absorb from the blackbody at infinity? This is the product of the surface area of the neutron star with the flux per unit area of the blackbody radiation.

A conundrum: compare the answer to (2) with the answer to (4). They differ. Does the neutron star cool down because (2) is greater than (4)?

The neutron star can't cool down because it is already in equilibrium, so one of our assumptions must be wrong.

5. It turns out that the most innocuous sounding assumption is incorrect. The power that the neutron star absorbs is the product of its apparent surface area with the flux per unit area of the blackbody radiation. Let the apparent radius be R_∞ and recalculate the answer to (4).
6. Equate (2) and (5) and solve for R_∞.

R_∞ ≠ R because in the vicinity of a neutron star light does not travel in a straight line. One can also derive the value of R_∞ by solving for a null geodesic that is tangent to the surface of the neutron star.

7. What is the minimum value of R_∞ for a constant value of M? You will need to know that

   1+z =

   1 (1 - 2 G M/R c2)1/2

   What is the value of R? Call this radius R_γ.
8. Prove the size of the image of the neutron star must decrease or remain the same as the radius of the neutron star decreases. Use the fact that rays that we ultimately see remain outgoing throughout their journey to us (otherwise by symmetry they would hit the surface a second time).
9. What happens to the size of the image of the star if the radius of the star is less that R_γ?
10. The calculation of the apparent radius of the star from thermodynamics hinges on the assumption that the outgoing flux from the surface reaches infinity. For R < R_γ, the size of the image no longer increases while thermodynamics says it should, so we must conclude that for radii less than R_γ initially outgoing photons can become incoming photons.

    From these arguments and spherical symmetry speculate what might happen to a photon emitted precisely at R_γ tangentially, i.e. neither ingoing or outgoing.

11. Calculate how much radiation a star whose radius is less than R_γ will absorb.

The answer to (11) falls short of (2) again. We know that the star can't heat up, so an assumption must be wrong. Within R_γ not every photon emitted can escape to infinity, many photons return and hit the surface.

12. Using the answers to (2) and (11), calculate the fraction of the outgoing photon flux emitted from the surface that manages to escape.
13. Use the fact that a blackbody emits isotropically to determine the opening angle of the cone into which the escaping photons are emitted. This region is symmetric around the radial direction.
14. Pat yourself on the back. You have derived many of the quirky things about the Schwarschild metric (the metric that surrounds a spherically symmetric mass distribution). List the key assumptions that you have made to make this derivation work.

Problem 2 - Light-Envelope Neutron-Star Cooling

The presence of hydrogen in the atmosphere of a neutron star strongly affects the emission of the surface of the star and possibly how the star cools.

1. Calculate the maximum possible density of pure degenerate ionized hydrogen gas (non-degenerate protons and degenerate electrons). What happens above this density?
2. Assume that the electrons dominate the pressure, what is the pressure at this critical density?
3. Assume that the gravitational acceleration is constant in this thin layer and is given by g_s,1410¹⁴ cm s^-2. What is the column density of the layer?
4. Assume that the stellar radius is R₆ 10⁶ cm. How much hydrogen can you pile onto a neutron star?
5. Assume that the cross section to x-rays per electron-proton pair is given by the Thomson cross-section. What is the column density of an optically thick layer of hydrogen (τ=1)?
6. The conductivity of the surface layers of a neutron star is inversely proportional to the atomic number of the nuclei. How does the surface luminosity of neutron star change if you pour hydrogen onto its surface?

Week 5 - Accreting Neutron Stars

Problem 1 - Accretion

1. Let's use Newtonian gravity for simplicity here. How much kinetic energy does a gram of material have if it falls freely from infinity to the surface of a star of mass M and radius R?
2. How much energy is released if a gram of material falls from a circular orbit just above the stellar surface onto the stellar surface? To put it another way, what is the kinetic energy of the material in the circular orbit?
3. Hydrogen burning releases about 6 x 10¹⁸ erg/g. How does accretion of hydrogen onto a neutron star (R=10km, M=1.4) differ from accretion onto a white dwarf (R=10000 km, M=0.6)?
4. What is the total about of energy released per gram of material as it falls from infinity to the surface of a neutron star? How many grams of material would have to fall each second on the neutron star to generate an Eddington luminosity through accretion? This is called the Eddington accretion rate.

Problem 2 - Bursts

We will try to model Type-I X-ray bursts using a simple model for the instability. We will calculate how much material will accumulate on a neutron star before it bursts.

1. Let us assume that the star accretes pure helium, that the temperature of the degenerate layer is constant down to the core (T_c), how much luminosity emerges from the surface of the star? (You shouldn't have to derive this formula (I gave it to you in class).
2. Let us assume that the helium layer has a mass, dM, and that the enregy generation rate for helium burning is given by

   ε3α = 3.5 x 1020 T9-3 exp(-4.32/T9) erg s-1 g-1

   where T₉=T/10⁹K. The energy generation rate is a function of density too, but let's forget about that to keep things simple. How much power does the helium layer generate as a function of dM?
3. Equate your answer to (1) to the answer to (2) and solve for dM. This is the thickness of a layer in thermal equilibrium.
4. Let's assume that the potential burst starts by the temperature in the accreted layer jiggling up by a wee bit. If the surface luminosity increases faster with temperature than the helium burning rate, then the layer is stable. Calculate dL_surface/dT and dP_helium/dT.
5. Calculate the value of dM for which dP_helium/dT exceeds dL_surface/dT and the layer bursts.
6. Equate your value of dM in (3) and (5) and solve for T. What is dM? How long will it take for such a layer to accumulate if the star is accreting at one-tenth of the Eddington accretion rate?

Week 6 - Class Suggestion and Project Discussion

Week 7 - Millisecond Pulsars

Problem 1 - How much mass?

1. Let us suppose that the radius of the inner edge of the accretion disk is given by r. What is the orbital frequency (ω) at the inner edge of the disk?
2. Using Newtonian physics, what is the specific angular momemtum (the angular momemtum per unit mass) of material at the inner edge of the disk?
3. Let us assume that the moment of inertia of a neutron star is given by I=0.2 M R² and that R is constant as the mass of the star increases. What is the angular momentum of the star if its rotation frequency is Ω (M) ?
4. What is the derivative of the angular momentum of the star with respect to mass, assuming that Ω is a function of M?
5. Set dL/dM in part (3) equal to l/m in part (2) and solve for d&Omega / dM.
6. This equation actually has an analytic solution (you are welcome to use Maple to find out). However, we don't want to work so hard, so let's estimate the various terms. Ω varies from zero to 4000 Hz. Take M=1.4 and R=r=10 km.
7. You should find that one term in the numerator is much larger than the other. Neglect the smaller term and replace dΩ with ΔΩ = 4000 Hz and dM with ΔM. Solve for ΔM.
8. We replaced the differential equation with a difference equation that we could solve algebraically. This won't give a large error if ΔM << M. Is this the case?

Problem 2 - The Birth Line

1. What is the spin frequency of a neutron star in equilibrium with an accretion disk whose inner radius is r_A?
2. Substitute for r_A using the expression from the lectures that gives r_A in terms of μ, G, M, R and L.
3. Now let's pretend that the accretion has stopped so we can use the radio pulsar expression for μ = B_p R³ for dipole spin-down. Substitute this expression for μ into the answer for (2), substitute Ω = 2π/P and solve for P-dot in terms of P.
4. To make sense of this expression let M=m, L=1.3 (l m) x 10³⁸ erg s^-1, R=10⁶ R₆ cm, I=10⁴⁵I₄₅ g cm² and sinα = 1. Does your birth line agree with the line in the lectures or the Bhattacharya paper?

Week 8 - Black Holes

Problem 1 - Photon Orbit

1. Start with the Schwarschild metric. We want a circular orbit so we will set dr=0, d (theta)=0 and theta=π/2. What is ds² for a photon (a photon travels along a null geodesic)? Solve for (dφ/dt)².
2. dφ/dt is simply Ω for the photon orbit. Kepler's third law works in the Schwarschild spacetime for circular orbits. Solve for M.

Problem 2 - Kepler's Law

1. Let's suppose that the particle at one moment is just going around the center of the black hole so the velocities in the r and theta directions vanish and we'll take theta=π/2 (the equatorial plane).

   In this situtation u^t and u^φ are the only components of the four velocity that don't vanish and Γ^r_tt and Γ^r_φφ are the only Christoffel symbols that don't vanish. Write out the geodesic equations.

2. We would like for the velocity to be constant around the circular orbit so we would like the first term in the geodesic equation to vanish. Solve for Ω = u^φ/u^t in terms of the non-vanishing Christoffel symbols.
3. The two non-vanishing Christoffel symbols are
   

   &Gammartt =

   (r - 2 M) M r3

   and &Gamma^r_φφ = (2 M - r) sin² theta. What is Ω in terms of M and r?
4. Substitute your value of Ω into the Schwarzschild metric and calculate ds² along the circular orbit. Over what range of radii can a material object (a toaster, UBC undergrad etc.) travel in a circular orbit around a Schwarzschild black hole.

Week 9 - Stellar Black Holes

Problem 1 - A Simplified Accretion Disk

1. Let's divide the accretion disk into a series of rings each of mass dm. What is the total energy of a ring at a distance r from the central black hole of mass M?
2. Let's say that the ring shrinks by a distance dr. What is the change in the energy of the ring (dE/dr) ?
3. As the ring shrinks mass is moving toward the black hole. Divide both sides the answer to (2) by dt to get an equation for the energy loss rate per radial interval.
4. What is the energy loss rate per unit area?
5. Let's assume that this energy is radiated at the radius where it is liberated. Using the blackbody formula what is the temperature of the surface of the disk?
6. Let's assume that the disk extends from an outer radius r_A to an inner radius r₀. What is the total luminosity of the disk if the accretion rate is dm/dt? What and where is the peak temperature of the disk? What and where is the minimum temperature of the disk?
7. Sketch the spectrum from the accretion disk on a log-log plot. You can use temperature units for the energy axis (i.e. kT_max and kT_min). To do this you will have to think about the peak flux from a blackbody at a particular temperature and the size of the disk that radiates at T_max and T_min.
8. The accretion rate is determined by the evolution of the orbit of the black hole with its companion, so it doesn't know about the Eddington limit of the black hole. What do you suppose happens if the rate that matter falls onto the disk exceeds the Eddington limit?
9. What major bit of physics has been left out of this analysis?

Problem 2 - Thinking about Instruments

1. The black hole in the center of our Galaxy has a mass of 10⁶ . Let us assume that it is a maximally rotating (a=M) Kerr black hole. How big is its horizon? How big is its ergosphere?
2. What angle does the horizon of the central black hole subtend in the sky?
3. I would like to build a telescope that can resolve the central black hole. What is the angular resolution of a telescope as a function of the wavelength of the light and the diameter of telescope. You can look up the formula, use dimensional analysis or the Heisenberg uncertainty principle.
4. What is the diameter of the telescope if you use 2 GHz radio waves?
5. What is the diameter of the telescope if you use 1 keV X-ray photons? Scaling from Chandra, what is the focal length of the telescope?

Week 10 - Supermassive Black Holes

Problem 1 - Accretion-Disk Efficiency

1. Using Newtonian gravity, how much energy is released per unit mass as material spirals to the inner edge of the disk?
2. Now using general relativity, redo the calculation. Assume that the central object is a non-rotating black hole. The energy released per unit mass is given by 1-u_t where u^α is the four-velocity of material in the disk and you are using the Schwarzschild metric.
3. In general relativity, an accretion disk can only extend down to R=6M around a non-rotating black hole. What is the efficiency of accretion onto such a black hole?

Problem 2 - Holes v. Stars

It turns out that the masses of black holes in the centers of galaxies is well correlated with the mass of the bulge of the galaxy (if it is a spiral galaxy) or the entire galaxy if it is an elliptical: M_BH ≈ 0.016 M_bulge.

1. Let's take a bulge of 10⁸ . If the black hole was built up by accretion over the age of the universe, what would its average luminosity be? Let's assume that it is a Schwarzschild hole.
2. The mass-to-light ratio of the bulges of galaxies is given by

   Mb Lb

   = 0.0776

   / | \

   Lb

   \0.18 | /

   What is the luminosity of the stars in bulge?

Problem 3 - Our Very Own Supermassive Black Hole

Andrea Ghez's group at UCLA constructed this beautiful movie of the centralmost arcsecond of our Galaxy. The edge of the box measures on arcsecond on the sky.

1. Use the movie to estimate the mass of the black hole at the center of our Galaxy.
2. How many Schwarzschild radii does the closest star approach the black hole?
3. How big would the black hole look on the sky to the hapless inhabitants on a planet orbiting this star? Would it be as big as the moon, Jupiter, Mars?
4. You have probably assumed something about the orbit of one of the stars. What did you assume? How does the mass of the black hole change if you vary this assumption? What could Andrea's team do to tighten the estimate of the black hole mass?

Week 11 - The Gamma-Ray Burst Controversy

Problem 1 - Isotropy

1. Suppose that GRBs lie in a spherical distribution of radius R, and we lie a distance R₀ away from the center. Furthermore suppose that we can see every GRB in the volume and we have seen a total of N GRBs. How many GRBs will we have seen in the hemisphere of sky toward the center of the distribution versus the opposite hemisphere?
2. Using the results from number (1), what is the difference in the number of GRBs in each hemisphere to lowest order in R₀/R?
3. Let's compare this result with observations. The one-sigma error in the difference between number of objects in the hemispheres is given by approximately by N^1/2. If you have observed N objects without detecting an difference between the two hemispheres at the two-sigma level, what is the minimum acceptable value of R/R₀?
4. At the time of the Paczynski-Lamb debate BATSE had observed about 600 bursts and R₀ was assumed to be 10kpc, what was the minimal acceptable value of R at the two-sigma level? Does this distance sound familiar?

Problem 2 - Boosting

1. Use the Minkowski metric to figure this out. I measure a photon to have an energy E. What is the four-momentum of the photon?
2. My pal is travelling toward me in the opposite direction of the photon at a velocity β c. What is his four-velocity? Use the definition γ = (1-β²)^-1/2 to simplify the expression. What energy would he measure for the photon? What does the expression look like as γ gets much larger than one?
3. If my pal observes the photon to have an energy of 100 MeV while I say its energy is less than 500 keV, what is the minimal value of γ for my pal (take β=1 to make life easier)?
4. My pal is still coming toward me at a velocity β c. When he is a distance r away from me (at a time t₀) he emits a photon toward me. How long does it take this photon to reach me?
5. From his point of view a short time Δt later he emits another photon toward me. How long is Δt in my frame and when do I receive the second photon? What is the difference in time between when I receive the first and second photons? What does the expression look like as γ gets much larger than one? Compare it with you answer to (2).
6. Explain two ways that relativistic motion (big values of γ) can relieve the compactness problem.

Problem 3 - Fermi Process

1. He is still toward me at a high value of γ. He serves a photon with energy E in his frame toward me. What is its energy when I receive it (take the high γ limit)?
2. I reflect the photon back toward him with a mirror. What is the energy of the photon in my pal's frame when he receives it again? What is its energy after another complete volley? After n complete volleys?
3. Let's say that we're not so good and that the probability that we successfully reflect the photon is p where p is much smaller than unity. What is the probability of completing n complete volleys?
4. Take the logarithm of both sides of the energy of the photon after n complete volleys and solve for n.
5. Take the logarithm of both sides of the probability of completing n complete volleys and substitute the value of n from the previous part. Rearrange to obtain an expression for the probability as a function of the photon energy -- it should be a power-law.
6. Take p to be 10^-5 (we're not very good) and γ = 100. What is the exponent of the power-law?

Week 12 - Understanding Cosmological Gamma-Ray Bursts

Problem 1 - Burst Rates

1. Black holes: what fraction of stars has initial masses greater than 20 ?
2. Neutron stars: what fraction of stars has initial masses greater than 8 and less than 20 ?
3. In steady state, if a galaxy has a star-formation-rate of 1 /yr, what is the rate of supernova that produce black holes and neutron stars?
4. Binary neutron stars: let's assume that half of all stars are in binaries and that stars pick their companions randomly, what fraction of neutron stars will have companions that will become neutron stars?
5. Finally, let us assume that only half of binaries survive a supernova explosion, what is the rate of binary-neutron-star mergers in the galaxy?
6. What is the mean star-formation rate of the universe? Take Ω_*=0.00245. You want a round number. What is the rate of GRBs in the two models?

Problem 2 - Neutrino-Eddington Limit

1. Use the cross section for neutrino pair-production as an estimate of the cross for a neutrino to scatter of an electron. What is the neutrino Eddington limit to the luminosity as a function of the mass of the star in solar masses and the energy of the neutrinos?
2. What is the Eddington limit to the accretion rate?
3. Use this Eddington-limited accretion rate to estimate the maximum value of Γ for a gamma-ray burst.

Problem 3 - The Millisecond Magnetar

1. What is P-dot for the magnetar when it is born?
2. What is the initial spin-down luminosity of the magnetar?
3. Does the spin-down luminosity of the magnetar increase or decrease with time? What does this mean in the context of the internal shocks model for gamma-ray burst emission?

Week 13 - Gamma-Ray Burst Afterglows